Lemma 10.32.5 (00IM)—The Stacks project

Table of contents
Part 1: Preliminaries
Chapter 10: Commutative Algebra
Section 10.32: Locally nilpotent ideals
Lemma 10.32.5 (cite)

An ideal in a Noetherian ring is nilpotent if each element of the ideal is nilpotent.

Lemma 10.32.5. Let $R$ be a Noetherian ring. Let $I, J$ be ideals of $R$. Suppose $J \subset \sqrt{I}$. Then $J^ n \subset I$ for some $n$. In particular, in a Noetherian ring the notions of “locally nilpotent ideal” and “nilpotent ideal” coincide.

Proof. Say $J = (f_1, \ldots , f_ s)$. By assumption $f_ i^{d_ i} \in I$. Take $n = d_1 + d_2 + \ldots + d_ s + 1$. $\square$

Comments (3)

Comment #851 by Bhargav Bhatt on July 26, 2014 at 03:22

Suggested slogan: An ideal in a noetherian ring is nilpotent if each element of the ideal is nilpotent.

Comment #9440 by Ryo Suzuki on June 10, 2024 at 18:26

It can be slightly generalized as follows: Let $R$ be a ring. Let $I$ be an ideal of $R$ and $J$ be a finitely generated ideal of $R$ . Suppose $J\subset I$ . Then $J^n\subset I$ for some $n$ .

Then Lemma 00L6 also can be generalized in the same way.

Comment #9441 by Ryo Suzuki on June 10, 2024 at 18:27

Oops! I rewrite it.

Let $R$ be a ring. Let $I$ be an ideal of $R$ and $J$ be a finitely generated ideal of $R$ . Suppose $J\subset \sqrt I$ . Then $J^n\subset I$ for some $n$ .

There are also:

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